## Calculus with Applications (10th Edition)

$$f''\left( x \right) = - \frac{6}{{{x^{5/4}}}},f''\left( 0 \right) = {\text{Undefined}},f''\left( 2 \right) = - \frac{6}{{{2^{5/4}}}}$$
\eqalign{ & f\left( x \right) = 32{x^{3/4}} \cr & {\text{find the derivative of }}f\left( x \right) \cr & {\text{ }}f'\left( x \right) = \frac{d}{{dx}}\left[ {32{x^{3/4}}} \right] \cr & {\text{ }}f'\left( x \right) = 32\frac{d}{{dx}}\left[ {{x^{3/4}}} \right] \cr & {\text{use power rule }}\frac{d}{{dx}}\left[ {{x^n}} \right] = n{x^{n - 1}} \cr & {\text{ }}f'\left( x \right) = 32\left( {\frac{3}{4}{x^{ - 1/4}}} \right) \cr & {\text{ }}f'\left( x \right) = 24{x^{ - 1/4}} \cr & \cr & {\text{find the derivative of }}f'\left( x \right) \cr & f''\left( x \right) = \frac{d}{{dx}}\left[ {24{x^{ - 1/4}}} \right] \cr & f''\left( x \right) = 24\frac{d}{{dx}}\left[ {{x^{ - 1/4}}} \right] \cr & {\text{use power rule}} \cr & f''\left( x \right) = 24\left( { - \frac{1}{4}{x^{ - 5/4}}} \right) \cr & f''\left( x \right) = - 6{x^{ - 5/4}} \cr & f''\left( x \right) = - \frac{6}{{{x^{5/4}}}} \cr & \cr & {\text{find }}f''\left( 0 \right){\text{ and }}f''\left( 2 \right) \cr & f''\left( 0 \right) = - \frac{6}{{{{\left( 0 \right)}^{5/4}}}} = \frac{6}{0}.{\text{ then the second derivative is not defined for }}x = 0 \cr & f''\left( 2 \right) = - \frac{6}{{{{\left( 2 \right)}^{5/4}}}} = - \frac{6}{{{2^{5/4}}}} \cr}